RSA Encryption Cryptography

RSA was one of the first public-key* cryptosystems and it is widely used for secure data transmission. It was first created by Ron Rivest, Adi Shamir, and Leona Adleman.

*Public key is used to establish a secret key, and the public key is sent in public. We then use the private key method to encrypt and decrypt large amounts of data, but no one knows the private key.

To code: U^s=x X(mod N)=Y

To decode Y^t=O O(mod N)=U

In computing, the modulo operation finds the remainder after division of one number by another. Given two positive numbers, a and n, a modulo n (in other words a mod n) is the remainder of the a division of a by n, where a is the dividend and n is the divisor.

Modular Arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers “wrap around” upon reaching a certain value—the modulus (plural modulae). We also had to deal with a lot of prime numbers, which is a whole number greater than 1 whose only factors are 1 and itself. A factor is a whole number that can be divided evenly into another number. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Numbers that have more than two factors are called composite numbers. the reason for covering primes was so that when we came to the coding part we actually had a s and t (what those are will be covered later) and more specifically relatively prime numbers – two numbers that don’t share any factors other than one, but aren’t prime themselves!!!!

Now to explain a bit of coding….

Let’s say we have this list 28, 9, 32, 33, 19, 0, 20, 1, 33, 11, 0, 27, 1, 20, 22 how would we decode it? Well for starters you need the formula Y^t=O O(mod N)=U where y is the original number so start from the beginning of that list and take the number so in this case its 28 then you need to take that to the power of t which is 7 at the moment. and your answer is O which is 13,492,928,512. Then take that large number and do that mod n which equals 7.And now find what letter 7 represents on the cypher chart. As shown it represents g. 28^7(mod 35)=7

Then you continue going from there so the next one is

9^7(mod 35)=9,

which corresponds to the letter i.

Side Note: Not every cypher is going to line up the same way, although in our problem we are using the cypher where a=1, b=2, etc.

Our Personal Experience:

Although the math part was a little difficult, we realized that we learned much better with someone showing us the work, rather than reading it from the packet. We also learned to ask for help a lot. There was a bit of struggle figuring out what this ≡ symbol means; we eventually realized it is used to denote both equality by definition (e.g., means is defined to be equal to) and congruence (e.g., means 13 divided by 12 leaves a remainder of 1 — a fact known to all readers of analog clocks).